Potts model calculation device

ABSTRACT

A Potts model computing device capable of computing a Potts problem that is a multivalued spin problem are described herein. The Potts model computing device includes: an Ising model computing device; a computation result storage and determination unit configured to store a value of a spin of the Ising model obtained in a case where a coupling coefficient is set in the Ising model computing device and to determine whether a computation is finished; and a coupling coefficient overwriting unit configured to update a coupling coefficient generated based on the stored value of the spin to the Ising model computing device. According to a value of a set of spins obtained as a computation result corresponding to a coupling coefficient set for an m-th time in the Ising model computing device, the coupling coefficient overwriting unit generates again a coupling coefficient to be set for an (m+1)-th iterative computation.

TECHNICAL FIELD

The present invention relates to a Potts model computing device whichsimulates the Potts model using optical pulses.

BACKGROUND ART

A conventionally-known von Neumann computer cannot efficiently solve acombinatorial optimization problem classified as an NP problem. As amethod of solving a combinatorial optimization problem, there isproposed a method using the Ising model, which is a lattice model for astatistical mechanics analysis of a magnetic material as an interactionbetween spins arranged in respective sites of a lattice.

It is known that a Hamiltonian H, which is an energy function in theIsing model, is expressed by the following formula (1).H=Σ _(ij) I _(ij)σ_(i)σ_(j)  (1)

Here, J_(ij) is a coupling coefficient indicating a correlation betweenspins on I and j th sites. σ_(i) and σ_(j) represent spins of the sitesand each take on a value of 1 or −1.

In the case of using the Ising model to solve a combinatorialoptimization problem, an optimum solution can be attained by obtainingσ_(i) at which the system is in a stable (ground) state and the value ofenergy H is the lowest when the Hamiltonian of the Ising model isprovided with J_(ij), the interaction between the sites. In recentyears, attention is being given to a computing device capable of solvinga combinatorial optimization problem such as an NP problem by simulatingthe Ising model using optical pulses (see PTL 1 and NPL 1).

FIG. 1 is a diagram showing a basic configuration of an Ising modelcomputing device. As shown in FIG. 1, the Ising model computing deviceis configured to generate a train of optical pulses corresponding to thenumber of sites of the Ising model by injecting a pump optical pulse(pump) into a phase sensitive amplifier (PSA) 2 provided in aring-shaped optical fiber which functions as a ring resonator 1 (binaryoptical parametric oscillator [OPO]; 0 or π phase optical parametricoscillator). If the optical pulse train input to the ring resonator 1circulates to reach the PSA 2 again, the pump light is input to the PSA2 again, whereby the optical pulse train is amplified. The optical pulsetrain generated by the first injection of the pump light includes weakpulses with an unstable phase, and is amplified by the PSA 2 each timeit circulates through the ring resonator 1, whereby the phase state isgradually determined. Since the PSA 2 amplifies each optical pulse inthe 0 or π phase with respect to the phase of the pump light source, thepulse is determined in either of the phases.

The Ising model computing device implements 1 and −1 of the spins in theIsing model in association with the 0 and π phases of the opticalpulses. The phases and amplitudes of the optical pulse train aremeasured by a measurement unit 3 outside the ring resonator 1 each timethe optical pulses circulate, and the measurement result is input to anarithmetic unit 4 provided with the coupling coefficient J_(ij) inadvance and is used to compute a coupling signal (feedback input signal)for the i-th pulse,Σ_(j) J _(ij) c _(j)(c_(j) is the amplitude of an optical pulse in the j-th site).

Further, the computed coupling signal is used to make a correlationbetween optical pulses forming the optical pulse train by generating anexternal optical pulse according to the coupling signal computed by anexternal optical pulse input unit 5 and inputting the external opticalpulse to the ring resonator 1.

The Ising model computing device can obtain a solution of the Isingmodel by circulating and amplifying the optical pulse train in the ringresonator 1 while providing the correlation described above, andmeasuring the 0 and π phases of the respective optical pulses formingthe optical pulse train in a stable state.

CITATION LIST Patent Literature

-   PTL 1: International Publication No. WO 2015/156126

Non Patent Literature

-   NPL 1: T. Inagaki, Y. Haribara, et al., “A coherent Ising machine    for 2000-node optimization problems,” Science 354, 603-606 (2016)-   NPL 2: Wu, Fa-Yueh (1982), “The Potts model,” Rev. Mod. Phys. 54    (1): 235-268

SUMMARY OF INVENTION Technical Problem

Incidentally, a solution to an Ising problem that can be solved by theIsing model computing device is either of binary spins. However, thereare a number of combinatorial optimization problems in which a spin tobe a solution takes one of more than two values (multiple values),so-called Potts problems (multivalued spin problems). Multivalued Pottsproblems with more than two values cannot be directly solved by theIsing model computing device, which deals with binary problems. However,since Potts problems that are multivalued problems are widelyapplicable, there has been a need for a device capable of computing thePotts model (NPL 2) for solving multivalued combinational optimizationproblems.

The present invention has been accomplished in view of the conventionalproblem described above. An object of the present invention is toprovide a Potts model computing device capable of computing a Pottsproblem that is a multivalued spin problem.

Solution to Problem

To achieve the object, the invention according to one embodiment is aPotts model computing device comprising: A Potts model computing devicecomprising: an Ising model computing device; a computation resultstorage and determination unit configured to store a value of a set ofspins of the Ising model obtained in a case where a coupling coefficientis set in the Ising model computing device and to determine whether acomputation is finished; and a coupling coefficient overwriting unitconfigured to update a coupling coefficient generated based on thestored value of the set of spins to the Ising model computing device,wherein, according to values of the spins of the Ising model obtained asa computation result for an m-th time iterative computation using theIsing model computing device, the coupling coefficient overwriting unitgenerates a coupling coefficient to be set for an (m+1)-th timecomputation and updates the generated coupling coefficient to the Isingmodel computing device, the computation result storage and determinationunit determines that a computation is finished in a case where a numberof iteration times reaches to Ms (Ms is a natural number), and computesa value S_(i) by substituting a value σ_(i m) of a spin obtained as acomputation result of the m-th time iterative Ising model computationinto the following formula to compute a problem mapped to the Pottsmodel using the Ising model computing devices:S _(i)=Σ_(m=1) ^(Ms)(1+σ_(im))2^(m−2)wherein a possible value of a multivalued spin of the Potts model isS_(i)=0, 1, 2, . . . , M−1 (M is a natural number) and M≤2^(M S).

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a diagram showing a basic configuration of a conventionalIsing model computing device.

FIG. 2A is a diagram in which a problem of coloring Kyushu with eightcolors using respective prefectures as nodes is replaced with a Pottsproblem as an example.

FIG. 2B is a diagram showing the Potts problem of FIG. 2A separated intothree-layer graph problems.

FIG. 2C is a diagram showing a relationship between the three-layergraph problems and a Potts interaction.

FIG. 3 is a diagram showing a schematic configuration of an Ising modelcomputing device of the present embodiment.

FIG. 4 is a diagram showing a configuration example of a balancedhomodyne detector.

FIG. 5 is a flowchart of processing in the basic configuration of theIsing model computing device.

FIG. 6 is a diagram showing a schematic configuration of a Potts modelcomputing device.

FIG. 7A is a diagram showing a virtual node defined for implementing aproblem of a multivalued value Si not satisfying M=2^(M s) in the Pottsmodel computing device.

FIG. 7B is a diagram in which the virtual node defined in FIG. 7A isused to apply colors.

FIG. 7C is a diagram expressing FIG. 7B by layers.

DESCRIPTION OF EMBODIMENTS

Embodiments of the present invention will be described in detail.

Potts Model

A Potts problem (multivalued spin problem) dealt with in a Potts modelcomputing device of the present embodiment can be solved using theHamiltonian of the following formula (2) called Potts model. In theformula (2), a value of a multivalued spin in a site (node) i formingthe Potts model is represented by S_(i) (S_(i)=0, 1, 2, 3 . . . , M−1; Mis a natural number indicating a multivalued number), and a couplingcoefficient indicating a correlation between sites (nodes) forming thePotts model is represented by J_(ij) (i, j=1, 2, . . . N; indicesindicating sites; N indicates the number of sites). The Kronecker deltaδ(S_(i)−S_(j)) is 1 only when S_(i)=S_(i) and is 0 in other cases.H _(Potts)=Σ_(ij) J _(ij)δ(S _(i) −S _(j))  (2)

In the case of solving the Potts problem, a problem to be solved ismapped to J_(ij) in the formula (2). When the multivalued number can beexpressed as M=2^(M S), the formula (2) describing a multivalued spinproblem can be rewritten as the following formula (3) by using binaryIsing spins σ^(i m), σ_(j m)=(±1) with a degree ranging from 1 to Ms.σ_(j m) indicates a binary spin value in the site j whose interactionwith the site i having a binary spin value σ_(i m) with the degree m isconsidered.

$\begin{matrix}{H_{Potts} = {\Sigma_{ij}J_{ij}{\Pi_{m = 1}^{M_{s}}\left( \frac{1 + {\sigma_{im}\sigma_{jm}}}{2} \right)}}} & (3)\end{matrix}$

The product of the binary spin interactions in the formula (3) can berewritten as shown by the following formula (4).

$\begin{matrix}{H_{Potts} = {\Sigma_{ij}{J_{ij}\left( \frac{1 + {\sigma_{i1}\sigma_{j1}}}{2} \right)}\left( \frac{1 + {\sigma_{i2}\sigma_{j2}}}{2} \right)\left( \frac{1 + {\sigma_{i3}\sigma_{j3}}}{2} \right)}} & (4)\end{matrix}$

The formula (4) shows that (1+σ_(im)σ_(jm))/2 in each of the m-th orderterms is equivalent to a binary Ising problem expressed by the formula(1) except for the addition and multiplication of constants (anyconstants do not affect a solution of the Ising problem). That is, thePotts problem of the multivalued number M (=2^(M S)) can be representedby the product of Ms binary Ising problems. More specifically, in orderto derive an interaction between multivalued spins (Potts interaction),it is necessary to consider the Ms product of the Ising interactions(first interaction) between binary spins σ_(i m), σ_(j m) in each of them-th order terms as well as an interaction (second interaction) providedby the infinite product of the first interactions. At this time, thevalue S_(i) of the multivalued spin having the Potts interaction can beexpressed by the following formula (5) using the Ms m-th order binaryspins σ_(i m) in the formulae (3) and (4).S _(i)=Σ_(m=1) ^(M) ^(S) (1+σ_(im))2^(m−2)  (5)

Accordingly, the value of the multivalued spin S_(i) to be solved in thePotts problem can be obtained by obtaining the value of the m-th orderbinary spin σ_(i m) when the Ising-like Hamiltonian in the formulae (3)or (4) is brought into a stable state. Further, according to theformulae (3) and (4), the multivalued spin can be determined byobtaining Ms sets of values of N binary spins σ_(i m), σ_(j m) (i, j=1,2, . . . N; indices indicating sites; N represents the number of sites).That is, a problem mapped to the Potts model can be computed bysubstituting (simulating) the binary spins σ_(i m), σ_(j m) (i, j=1, 2,. . . N) with phases of optical pulses (such as 0, π) and providing thephases of the optical pulses with the Potts interaction.

This transformation for the Potts model Hamiltonian will be furtherdescribed by giving an example of a map coloring problem. ColoringKyushu with eight colors can be regarded as a multivalued problem inwhich values from 0 to 7 are assigned to the respective colors used forcoloring. The problem can be expressed by using three sets of binaryspins σ_(i m), σ_(j m), where M=8=2³, that is, m=1, 2, 3 (i, j=1, 2, . .. N; indices indicating sites; N represents the number of sites).

FIG. 2A, FIG. 2B, and FIG. 2C are diagrams illustrating the Pottsinteraction in the problem of coloring Kyushu with eight colors usingrespective prefectures as sites shown as an example. The PottsHamiltonian for this problem with the three sets of binary spinsσ_(i m), σ_(j m) have an interaction can be described by the graph shownin FIG. 2B in which the original graph of Kyushu shown in FIG. 2A iscopied into three layers in order to consider the Potts interaction. Inthis coloring problem, whether two sites (prefectures) are adjacent toeach other is set to J_(ij) as a correlation. This becomes a problem tobe solved. For example, on the assumption that Nagasaki Prefecture isi=1 and Saga Prefecture is j=2, since these two prefectures are adjacentto each other, J₁ ₂=1 indicating adjacency is set as a correlation. InFIG. 2A and FIG. 2C, Nagasaki and Saga are connected by a thick line,which indicates the correlation.

First, two specific adjacent sites in each layer in the graph shown inFIG. 2B will be considered. It is determined that there is aninteraction if the values of spins of two sites on each layer are equalto each other on all of three layers and it is determined that there isno interaction if spins of two sites on each layers are different fromeach other. The presence/absence of an interaction between the twoadjacent nodes is determined in each of the three layer. If it isdetermined that “there is no interaction” in any one of the layers, itis determined that there is no Potts interaction between the two nodes.That is, it can be determined that there is a Potts interaction if thevalues of the adjacent nodes are equal to each other in all the layersas shown in FIG. 2C.

In the eight color coloring problem, each site can take on eight values.However, adjacent sites should be colored with different colors. Inother words, as to adjacent sites, a solution to the problem is a set ofmultivalued spins S_(i) obtained by substituting the values (±1) of thesites in each layer into formula (5) in the case where Potts interactionis absent on whole sites.

The multivalued problem can be solved by substituting (emulating) thevalues (±1) in each layer in the graph of multiple layers described inthis problem with phases of optical pulses (such as 0, π).

First Embodiment

FIG. 3 is a diagram showing a schematic configuration of a Potts modelcomputing device of the present embodiment. In FIG. 3, the Potts modelcomputing device has a ring resonator 1 formed by a ring-shaped opticalfiber, a phase sensitive amplifier (PSA) 2 provided in the ringresonator 1, a measurement unit 3, an arithmetic unit 4, and an externaloptical pulse input unit 5 forming part of a feedback loop branchingfrom the ring resonator 1, a computation result storage anddetermination unit 6 which stores a computation result corresponding toa coupling coefficient and determines whether a computation is finished,and a coupling coefficient overwriting unit 7 which sets a couplingcoefficient by updating the value of J_(ij) written to the arithmeticunit 4 according to the measurement unit 3. In the Potts model computingdevice of the present embodiment, a feedback loop is formed by themeasurement unit 3, the arithmetic unit 4, and the external opticalpulse input unit 5. In this configuration, a so-called Ising modelcomputing device is formed by the ring resonator 1, the PSA 2, themeasurement unit 3, the arithmetic unit 4, and the external opticalpulse input unit 5. The computation result storage and determinationunit 6 and the coupling coefficient overwriting unit 7 which can beformed by, for example, an external computer having a CPU and a storageunit such as a memory.

The PSA 2 efficiently amplifies light of the 0 or π phase with respectto the phase of the pump light source (to be exact, local light used forpump light pulse generation) out of a train of optical pulses having thesame oscillation frequency which simulates a spin set of the Isingmodel. The PSA 2 can be formed by using a nonlinear optical crystal suchas periodically poled lithium niobate (PPLN) having a second-ordernonlinear optical effect.

If signal light and pump light (excitation light) are input, the PSA 2generates a weak pulse (idler light) in the 0 or π phase with respect tothe phase of the pump light source. Even if only pump light is firstinput before generation of signal light, the PSA 2 can generate a weakpulse (noise optical pulse) as spontaneous emission light.

If pump light obtained by converting local oscillator light (LO light)having a frequency ω into second harmonic light having a frequency 2ω bya second harmonic generator is input (if there has been no pump lightand input of pump is just starting), the PSA 2 generates weak noiselight through a parametric down-conversion process. Further, if anoptical pulse train that has circulated and propagated through the ringresonator 1 is input to the PSA 2 again, the optical pulse train becomessignal light,

E_(s) = A_(s)e^(i(ω_(s)t + θ_(s)))and if the pump light completely phase-matched to the signal light,

E_(p) = A_(p)e^(iω_(p)t)is further input to the PSA 2, idler light to be phase conjugate lightof the signal light Es,

E_(i) = A_(i)e^(i{(ω_(p) − ω_(s))t − θ_(s)})is generated by optical parametric oscillation (OPO), which is thesecond-order nonlinear optical effect.

At this time, if the frequency of the signal light corresponds to thatof the idler light, the following degenerate wave is output.

E_(s) = A_(s)e^(i(ω_(s)t + θ_(s))) + A_(s)e^(i(ω_(s)t − θ_(s))) = 2A_(s)e^(iω_(s)t)cos θ_(s)

Since the output degenerate wave is obtained by addition of the signallight and the idler light in the phase conjugation relationship, a 0 orπ phase wave is efficiently amplified. Thus, the PSA 2 amplifies 0 or πphase components of the weak optical pulse train generated first.

The ring resonator 1 causes a plurality of optical pulses (optical pulsetrain) generated in the PSA 2 to circulate and propagate. The ringresonator 1 can be formed by a ring-shaped optical fiber. The length ofthe optical fiber is set at a value obtained by adding a lengthcorresponding to a time required for feedback processing to (the numberof pulses forming the optical pulse train)×(a pulse interval).

The measurement unit 3 functions as an optical pulse measurement unitthat measures the phases and amplitudes of a plurality of optical pulses(optical pulse train) each time the optical pulses circulate through thering resonator 1 (for each circulation). More specifically, themeasurement unit 3 causes the optical pulse train propagating throughthe ring resonator 1 to branch and makes a coherent measurement of thephase state including amplitudes. The coherent measurement can measurethe amplitudes and phases of the optical pulse train input as light tobe measured by means of a balanced homodyne detector.

FIG. 4 is a diagram showing a configuration example of a balancedhomodyne detector 30. The balanced homodyne detector 30 can set, asreference light, phase-locked light having the same frequency as anoptical pulse train to be measured, cause the reference light tointerfere with light forming the optical pulse train, and measure theamplitude and phase state of the optical pulse train. The balancedhomodyne detector 30 has a half mirror 31 which causes light from a port1 and light from a port 2 to interfere with each other and outputs themto a port 3 and a port 4, a first photodetector 32 which detects lightoutput from the port 3, a second photodetector 33 which detects lightoutput from the port 4, and a difference computing unit 34 whichcomputes a difference between the detection results of the first andsecond photodetectors 32 and 33.

An optical pulse train E_(s)e^(i(ωt+0)) is input to the port 1 as lightto be measured and reference light E_(L o) e^(iωt) whose amplitude andphase are known is input to the port 2. The optical pulse train input tothe port 1 branches at the half mirror 31 into an in-phase componenttransmitted toward the port 3 and a component reflected toward the port4 with a phase changed by π. The reference light input to the port 2branches at the half mirror 31 into an in-phase component transmittedtoward the port 4 and an in-phase component reflected toward the port 3.

Output light obtained by interference between the in-phase component ofthe optical pulse train input to the port 1 and the in-phase componentof the reference light input to the port 2,

$E_{3} = {{\frac{E_{L0}}{\sqrt{2}}e^{i\omega t}} + {\frac{E_{s}}{\sqrt{2}}e^{i({{\omega t} + \theta})}}}$is output from the port 3. The first photodetector 32 detects anelectric signal indicating light intensity,

${❘E_{3}❘}^{2} = {\frac{E_{LO}^{2} + E_{S}^{2}}{2} + {E_{LO}E_{S}\cos\theta}}$

The opposed phase component of the optical pulse train input to the port1 interferes with the in-phase component of the reference light input tothe port 2, whereby output light,

$E_{4} = {{\frac{E_{LO}}{\sqrt{2}}e^{i\omega t}} - {\frac{E_{S}}{\sqrt{2}}e^{i({{\omega t} + \theta})}}}$

is output from the port 4. The second photodetector 33 detects anelectric signal expressed by light intensity,

${❘E_{4}❘}^{2} = {\frac{E_{LO}^{2} + E_{S}^{2}}{2} - {E_{LO}E_{S}\cos\theta}}$

The difference computing unit 34 computes a difference between thedetection signal in the first photodetector 32 and the detection signalin the second photodetector 33 to output 2E_(L o) E_(s) cos θ.

Since the amplitude E_(L o) of the reference light is known, a value ±Eincluding a cosine (in-phase) component (sign only) of the phase and anamplitude is obtained as a measurement result.

The value obtained as a measurement result is an analog value with asign (±E). The sign (±) indicates the phase (0 and π) and the analogvalue (E) indicates the amplitude.

Returning to FIG. 3, the arithmetic unit 4 computes an interactionbetween optical pulses, using information about the phases andamplitudes of the measured optical pulses as an input, based on acoupling coefficient mapped to the Ising model and information about thephases and amplitudes of the other optical pulses. The measurement ofoptical pulses is done by the arithmetic unit 4 for only a specificcomponent (in-phase component) of the phases and amplitudes.

More specifically, the arithmetic unit 4 makes a computation accordingto the following formula (6) with the amplitudes and phases of theoptical pulse train measured by the measurement unit 3 and the couplingcoefficient. As the arithmetic unit 4, for example, an FPGA can be used.

$\begin{matrix}{\begin{pmatrix}f_{1} \\f_{2} \\f_{3} \\f_{4} \\f_{5}\end{pmatrix} = {\begin{pmatrix}0 & J_{12} & J_{13} & & J_{14} & J_{15} \\J_{21} & 0 & J_{23} & & J_{24} & J_{25} \\J_{31} & J_{32} & 0 & & J_{34} & J_{35} \\J_{41} & J_{42} & J_{43} & & 0 & J_{45} \\J_{51} & J_{52} & J_{53} & & J_{54} & 0\end{pmatrix}\begin{pmatrix}c_{1} \\c_{2} \\c_{3} \\c_{4} \\c_{5}\end{pmatrix}}} & (6)\end{matrix}$

In the above formula, c₁, c₂, c₃, c₄, c₅ indicate measurement results ofoptical pulses in the measurement unit 3 and f₁, f₂, f₃, f₄, f₅ indicateinteractions obtained as computation results. Arithmetic parameters J₁₂, J₁ ₃, J₁ ₄, J₁ ₅, J₅ ₃, J₅ ₄ of a matrix are coupling coefficientsthat describe an Ising model and are determined according to a problemto be solved. However, in the Potts model computing device of thepresent embodiment, the coupling coefficient is changed multiple timesbefore the provided multivalued problem is solved.

As shown by the above formula, the arithmetic unit 4 makes a computationof multiplying a matrix by a column vector including measurement resultsin the measurement unit 3, and obtains a column vector that describesinteractions between optical pulses. Although an example in which thenumber of sites equal to the number of optical pulses forming theoptical pulse train is five is described here, the size of the squarematrix to be used is determined according to the number of sites. Thesquare matrix has a size of (the number of sites)×(the number of sites).

For example, if the number of sites (the number of optical pulsesforming the optical pulse train) is N, an interaction computing unitmakes a computation of the matrix expressed by the following formula(7), wherein K_(ij)=J_(ij) ^(m),

$\begin{pmatrix}f_{1m} \\f_{2m} \\f_{3m} \\ \vdots \\f_{{({N - 1})}m} \\f_{Nm}\end{pmatrix} = {\begin{pmatrix}0 & K_{12} & K_{13} & \cdots & K_{1{({N - 1})}} & K_{1N} \\K_{21} & 0 & K_{23} & \cdots & K_{2{({N - 1})}} & K_{2N} \\K_{31} & K_{32} & 0 & \cdots & K_{3{({N - 1})}} & K_{3N} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\K_{{({N - 1})}1} & K_{{({N - 1})}2} & K_{{({N - 1})}3} & \cdots & 0 & K_{{({N - 1})}N} \\K_{N1} & K_{N2} & K_{N3} & \cdots & K_{N({N - 1})} & 0\end{pmatrix}{\begin{pmatrix}c_{1m} \\c_{2m} \\c_{3m} \\ \vdots \\c_{{({N - 1})}m} \\c_{Nm}\end{pmatrix}.}}$

The external optical pulse input unit 5 implements the magnitude andsign of an interaction relating to an optical pulse by controlling theamplitude and phase of an optical pulse to be overlapped on an opticalpulse in the ring resonator 1 by using the computation result computedbased on the phase and amplitude of an optical pulse in which only aspecific component of the phase and amplitude is measured. The externaloptical pulse input unit 5 can be formed, for example, using a laserwhich controls and outputs the amplitude and phase of an optical pulse.Since the computation result of the specific component of the phase andamplitude is obtained, an interaction of only the specific component ofthe phase and amplitude is implemented.

More specifically, the external optical pulse input unit 5 generates anexternal optical pulses having the same frequency as the optical pulsetrain in the ring resonator 1 in an amplitude and phase proportional tothe computation result. For example, the external optical pulse inputunit 5 synchronizes and inputs an external pulse of a constant frequencyso as to synchronize and input a pulse having a frequency correspondingto that of the optical pulse train in the ring resonator 1. The opticalpulse train overlapping with the optical pulse train in the ringresonator 1 yields interaction between the optical pulses in the ringresonator 1 that emulates the interaction between spins.

As described above, according to the configuration of feedback input bythe external optical pulse input unit 5, the signal c′_(i)(n) of theoptical pulse train after the feedback can be expressed by the followingformula (8) using the cosine (in-phase) component c_(i) of the i-thoptical pulse, the number n of circulations in the resonator, and therate K of the external pulse.c _(i)′(n)=c _(i)(n)+Kf _(i)(n)  (8)

In the above formula, the optical pulse train c′_(i)(n) after thefeedback is obtained by overlapping the optical pulse train c_(i)(n) inthe ring resonator 1 with the external optical pulse train (feedbackinput component),

$\sum\limits_{j}^{N}{J_{ij}{c_{j}(n)}}$by the external optical pulse input unit 5 at the coupling rate K.

When input to the PSA 2 again, the optical pulse train c′_(i)(n)expressed by the above formula is amplified to become an optical pulsetrain c_(i)(n+1). The above configuration enables the Potts modelcomputing device to repeatedly obtain a computation result correspondingto the coupling coefficient by bringing the optical pulse train into astable state according to the problem while repeating the amplificationand feedback.

When bringing the optical pulse train into a stable state according tothe problem while repeating the amplification and feedback apredetermined number of iteration times, the computation result storageand determination unit 6 stores a computation result of a set of binaryIsing spins obtained by substituting the phase of the optical pulsetrain with a spin value based on the measurement result measured by themeasurement unit 3, that is, (sign(c−_(i))=±1).

Further, the computation result storage and determination unit 6determines that a computation is finished if the amplification andfeedback in the above configuration and the computation for bringingabout a stable state (setting of the coupling coefficient in thecoupling coefficient overwriting unit 7) are repeated a number ofiteration times necessary for solving the provided Potts problem. If apossible multivalued value of spin is expressed by M=2^(Ms), the Pottsmodel computing device of the present embodiment may require Ms times ofiterative Ising model computations at minimum. In the end of the (Pottsmodel) computation, a multivalued spin value of the Potts model isfinally computed based on the computation results previously stored.

If a predetermined finish condition is satisfied, the computation resultstorage and determination unit 6 can finish the iteration of Ising modelcomputation even in the case where the number of iteration times ofcomputation is less than Ms. The predetermined finish condition isthat 1) the number of iteration reaches m=Ms, 2) a coupling coefficientgenerated subsequent to obtaining the m-th computation result is J_(ij)^(m+1), m+¹=0 (that is, any sign(c_(i m) c_(j m))=−1 in (i, j) whenJ_(ij) ^(m)≠0), or 3) a coupling coefficient generated subsequent toobtaining the m-th computation result is J_(ij) ^(m+1)=J_(ij) ^(m) (thatis, any sign(c_(i m) c_(j m))=1 in (i, j) when J_(ij) ^(m)≠0). When theabove condition is satisfied, the iteration of Ising model computationis finished even if the number of iterations is less than Ms. Even ifthe values of M and Ms for the coupling coefficient are undetermined,the values of M and Ms can be obtained by an algorithm for finishing acomputation based on the computation result.

The coupling coefficient overwriting unit 7 repeatedly sets the couplingcoefficient to the computing unit 3. More specifically, the couplingcoefficient overwriting unit 7 sets the first coupling coefficient atJ_(ij) ^(m=1)=J_(ij) and sets the second coupling coefficient onward(m+1-th coupling coefficient) at J_(ij) ^(m+1)=J_(ij) ^(m)×(1+sign(c_(i m) c_(j m)))×½, using a quadrature component c_(j m) of the i-thpulse and a quadrature component c_(j m) of the j-th pulse of the last(m-th) computation result. This setting makes it possible to regenerateJ_(ij) such that a correlation between nodes in which measured spinvalues do not correspond to each other is excluded from computationtargets.

After the coupling coefficient overwriting unit 7 sets the couplingcoefficient, a noise optical pulse train is generated again by the PSA 2to repeat the computation for bringing about a stable state. Thecomputation for bringing about a stable state is repeated Ms times, Msbeing a degree when the multivalued spin is expressed in binary (lessthan Ms times if the finish condition is satisfied), and the computationresult storage and determination unit 6 substitutes a set of spin valuesσ_(i m) obtained as computation results of the coupling coefficient setmultiple times into the formula (5) to obtain a value of multivaluedspins to be a solution to the Potts model.

FIG. 5 is a flowchart of processing in the basic configuration of thePotts model computing device. As shown in FIG. 5, in the Potts modelcomputing device, after the coupling coefficient overwriting unit 7 setsthe coupling coefficient (the number of iteration times of setting m=1)at the beginning of a computation of the Potts model, pump light isfirst injected to the PSA 2, followed by generation of a weak noiseoptical pulse train (S1). The generated noise optical pulse traincirculates and propagates through the ring resonator 1 and partiallybranches and the amplitudes and phases of the optical pulse train aresubjected to a coherent measurement by the measurement unit 3 (S2).

After the measurement result of the optical pulse train is obtained, thearithmetic unit 4 computes an interaction using a matrix to which thecoupling coefficient according to a problem to be solved is mapped (S4).Upon receipt of the computation result, the external optical pulse inputunit 5 inputs an external optical pulse generated based on thecomputation result to the ring resonator 1 and overlaps the externaloptical pulse with the optical pulse train in the ring resonator 1,thereby providing the optical pulse train with a feedback (S5).

The optical pulse train after the feedback is input again to the PSA 2,amplified by the pump light with the optical pulse train (S6), andcirculates and propagates again through the ring resonator 1. Thecoherent measurement, the computation using the matrix, and the feedbackaccording to the computation result are repeatedly performed for theoptical pulse train propagating again through the ring resonator 1.

If such an amplification and feedback for the optical pulse train arerepeated a predetermined number of iteration times (S3), the opticalpulse train is brought into a stable state. The computation resultstorage and determination unit 6 stores a computation result obtained bybringing about the stable state (obtaining the value of binary spins σ(=±1) from stable optical phase states 0 or π measured by themeasurement unit 3) (S7) and determines whether the number of iterationtimes of Ising model computation reaches to the degree Ms of the problemto be solved and whether the finish condition is satisfied (S8). If thenumber of iteration times does not reach Ms and the finish condition isnot satisfied (S8: No), the computation result storage and determinationunit 6 regenerates J_(ij) based on 0 or π, which is the phase state ofthe measurement result obtained by the measurement unit 3 in the stablestate, sets the generated J_(ij) to the arithmetic unit 4 (S9),generates a noise optical pulse train by the PSA 2 again (S1), andrepeats the measurement, feedback, and amplification to bring about astable state (repetition of S1 to S6, that is, repetition of Ising modelcomputation).

Finally, a solution to the provided problem can be obtained bysubstituting 0 or π, which is the phase state of the measurement resultobtained by the measurement unit 3 in the stable state when the numberof iteration times the state becomes stable corresponds to the degree ofthe problem to be solved or the finish condition is satisfied (S8: Yes),with the spin σ state (±1) of the Ising model and mapping it to theproblem to be solved again.

In the present embodiment, an example in which the Ising model computingdevice is formed by the ring resonator 1, the PSA 2, the measurementunit 3, the arithmetic unit 4, and the external optical pulse input unit5 has been described. However, the configuration of the Ising modelcomputing device is not limited to this.

Second Embodiment

FIG. 6 is a diagram showing a schematic configuration of a Potts modelcomputing device of the present embodiment. Although the Potts modelcomputing device of the first embodiment can directly deal with amultivalued number M expressed by 2^(M s), Ms power of 2 (Ms is anatural number), a multivalued number M not satisfying a formula such asM=3 cannot be directly implemented by the Potts model computing device.As shown in FIG. 6, the Potts model computing device of the presentembodiment introduces a virtual node having a correlation with all othernodes so as to have a configuration obtained by providing the couplingcoefficient overwriting unit 7 with an adjustment unit 71 in the Pottsmodel computing device of the first embodiment. This configurationenables a computation of a problem of a multivalued number notsatisfying M=2^(M s).

The adjustment unit 71 adjusts “the number of nodes,” “a possible valueof a multivalued spin on each node,” and “a correlation between nodes”of a provided problem and sets J_(ij) according to the adjusted problem.More specifically, if such a coupling coefficient that the number ofnodes is “N” and a multivalued spin value of each node is M is provided,the adjustment unit 71 determines that a possible value of each node is2^(M s) determined by the minimum Ms satisfying M≤2^(M s), defines thenumber of Dn virtual nodes (Dn=2^(M s)−M; M is a natural number), andadds Dn nodes to the original problem with N nodes such that the numberof nodes of the problem is (N+Dn). The adjustment unit 71 further mapsJ_(ij) on the assumption that a newly-introduced virtual nodes have acorrelation (coupling) with all the other nodes. For example, if a spinvalue of each node is different from a spin value of a node in thecoupling relationship, since the virtual node has a coupling to all theother nodes, it is clear that a possible value of the virtual nodeobtained as a solution is different from the values of all the othernodes. Accordingly, the rest of the multivalued values (multivaluedvalues originally set as a problem) are assigned to nodes other than thevirtual node.

FIG. 7A, FIG. 7B, and FIG. 7C are diagrams showing a method ofimplementing a problem of a multivalued value Si not satisfyingM=2^(M s) in the Potts model computing device. For example, in the caseof solving a coloring problem of coloring seven prefectures of Kyushu(N=7) with three colors, a multivalued number M is 3, which does notsatisfy M=2^(M s). Thus, the problem cannot be directly implemented inthe Potts model computing device of the first embodiment.

When a coloring problem of coloring seven prefectures of Kyushu (N=7)with three colors is input, since a possible value is 4 from M=3≤2² andthe number of virtual nodes is Dn=2²−3=1, the adjustment unit 71 definesone virtual node. The adjustment unit 71 generates J_(ij) on theassumption that the number of nodes is 8, which is larger by one (thenumber of virtual nodes) than the number of nodes input as a problem,and the number of colors to be used for coloring is 4, which is largerby one than the number of colors to be actually used, and maps thegenerated J_(ij) to the arithmetic unit 4. At this time, as shown inFIG. 7A, the correlation J_(ij) is generated based on the assumptionthat the virtual node is coupled to all the other nodes. As shown inFIG. 7C, the number of layers of graphs (the number of iteration timesof Ising model computation) necessary at this time is 2 from Ms=2.

By this setting, as shown in FIG. 7B, a color different from the colorsof the seven other nodes is guided to the virtual node. In other words,three colors are applied to the seven nodes to be colored.

The invention claimed is:
 1. A Potts model computing device comprising:an Ising model computing device; a computation result storage anddetermination unit configured to store a value of a set of spins of theIsing model obtained in a case where a coupling coefficient is set inthe Ising model computing device and to determine whether a computationis finished; and a coupling coefficient overwriting unit configured toupdate a coupling coefficient generated based on the stored value of theset of spins to the Ising model computing device, wherein, according tovalues of the spins of the Ising model obtained as a computation resultfor an m-th time iterative computation using the Ising model computingdevice, the coupling coefficient overwriting unit generates a couplingcoefficient to be set for an (m+1)-th time computation and updates thegenerated coupling coefficient to the Ising model computing device, thecomputation result storage and determination unit determines that acomputation is finished in a case where a number of iteration timesreaches to Ms (Ms is a natural number), and computes a value S_(i) bysubstituting a value σ_(i m) of a spin obtained as a computation resultof the m-th time iterative Ising model computation into the followingformula to compute a problem mapped to the Potts model using the Isingmodel computing devices:S _(i)=Σ_(m=1) ^(Ms)(1+σ_(im))2^(m−2) wherein a possible value of amultivalued spin of the Potts model is S_(i)=0, 1, 2, . . . , M−1 (M isa natural number) and M≤2^(M S).
 2. The Potts model computing deviceaccording to claim 1, wherein the Ising model computing devicecomprises: a phase sensitive amplifier configured to parametricallyoscillate an optical pulse train having a same oscillation frequency ina 0 or π phase, which emulates a set of binary spins of the Ising model;a ring resonator configured to allow the optical pulse train tocirculate and propagate; an optical pulse measurement unit configured tomeasure phases and amplitudes of the optical pulse train each time theoptical pulse train circulates and propagates through the ringresonator; an interaction computing unit configured to compute aninteraction between optical pulses using information about phases andamplitudes of optical pulses measured in the optical pulse measurementunit as input, the interaction being determined from the couplingcoefficient of the Ising model and the measured optical pulses; aninteraction implementation unit configured to implement the interactionbetween the optical pulse train determined based on the couplingcoefficient of the Ising model and the phases and amplitudes of themeasured optical pulses by controlling and overlapping amplitudes andphases of optical pulses on the optical pulse train in the interactioncomputing unit; and a problem overwriting unit configured to update thecoupling coefficient of the Ising model, in a process of repeatingfeedback loop control formed by the optical pulse measurement unit, theinteraction computing unit, and the interaction implementation unit, theoptical pulse measurement unit obtains a spin set of solution of theIsing model by converting the phases of the optical pulse train intobinary Ising spins after the optical pulse train reaches a stable state.3. The Potts model computing device according to claim 2, wherein theinteraction computing unit multiplies a column vector having phases andamplitudes c_(1 m), c_(2 m), c_(3 m), c_(4 m), c_(i m), c_((N−1)m),c_(N m) of N measured optical pulses as elements by the following matrixhaving coupling coefficients of the Ising model as arithmetic parametersand computes elements f_(1 m), f_(2 m), f_(3 m), f_(4 m), f_(i m),f_((N−1)m), f_(N m) of the obtained column vector Ms times asinteractions relating to N optical pulses corresponding to the N opticalpulses while varying m from 1 to Ms, wherein K_(ij)=J_(ij) ^(m):$\begin{pmatrix}f_{1m} \\f_{2m} \\f_{3m} \\ \vdots \\f_{{({N - 1})}m} \\f_{Nm}\end{pmatrix} = {\begin{pmatrix}0 & K_{12} & K_{13} & \cdots & K_{1{({N - 1})}} & K_{1N} \\K_{21} & 0 & K_{23} & \cdots & K_{2{({N - 1})}} & K_{2N} \\K_{31} & K_{32} & 0 & \cdots & K_{3{({N - 1})}} & K_{3N} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\K_{{({N - 1})}1} & K_{{({N - 1})}2} & K_{{({N - 1})}3} & \cdots & 0 & K_{{({N - 1})}N} \\K_{N1} & K_{N2} & K_{N3} & \cdots & K_{N({N - 1})} & 0\end{pmatrix}{\begin{pmatrix}c_{1m} \\c_{2m} \\c_{3m} \\ \vdots \\c_{{({N - 1})}m} \\c_{Nm}\end{pmatrix}.}}$
 4. The Potts model computing device of claim 1,wherein the computation result storage and determination unit determinesthat a computation is finished in a case where the computation resultstorage and determination unit determines that any one of the followingconditions is satisfied: 1) the Ising model computation is repeated Ntimes; 2) a coupling coefficient generated subsequent to obtaining them-th computation result is J_(ij) ^(m+1)=0; and 3) a couplingcoefficient generated subsequent to obtaining the m-th computationresult is J_(ij) ^(m+1)=J_(ij) ^(m).
 5. The Potts model computing deviceof claim 2, wherein the computation result storage and determinationunit determines that a computation is finished in a case where thecomputation result storage and determination unit determines that anyone of the following conditions is satisfied: 1) the Ising modelcomputation is repeated N times; 2) a coupling coefficient generatedsubsequent to obtaining the m-th computation result is J_(ij) ^(m+1)=0;and 3) a coupling coefficient generated subsequent to obtaining the m-thcomputation result is J_(ij) ^(m+1)=J_(ij) ^(m).
 6. The Potts modelcomputing device of claim 3, wherein the computation result storage anddetermination unit determines that a computation is finished in a casewhere the computation result storage and determination unit determinesthat any one of the following conditions is satisfied: 1) the Isingmodel computation is repeated N times; 2) a coupling coefficientgenerated subsequent to obtaining the m-th computation result is J_(ij)^(m+1)=0; and 3) a coupling coefficient generated subsequent toobtaining the m-th computation result is J_(ij) ^(m+1)=J_(ij) ^(m).